Question

# Question 1 This question concerns the volume of an object bounded by a parabola x = y² - 5y on the left and a straight line x = y on the right (z and y in metres). The cross section is show below: Ty x = y? — 5y The height of the object, h(x, y) (in metres), is modelled in this assessment using the following function: h(x,y) = (y − z)(z − y? +5y). x=y (a) (5 marks) Find the maximum height by doing the following: i) find all the critical points of the function h(x, y), ii) identify the single critical point that is within the cross section area A depicted above (not including points on the boundary). iii) show that this critical point is a local maximum using the Hessian determinant test, iv) evaluate the value of h at this maximum (b) (5 marks) The volume of the object V (metres³) is defined as the integral =ff h(x, y) da V = where A is the cross-section depicted in the figure above. Calculate the volume by doing the following:/n(c) (2 marks) use MATLAB to create a contour plot of the height h(x, y). Make sure to include a few (three or more) contours (level curves) between zero and the maximum value you found in Qla.  Fig: 1  Fig: 2